Generalized F-depth and graded nilpotent singularities

Abstract

In this article, we provide explicit and effective bounds for Frobenius test exponents, integers crucial for computing Frobenius closures of parameter ideals, in new settings and articulate their explicit computation along common graded constructions. We do this via the study of variants of F-nilpotent singularities. In particular, we explore how (generalized) weakly F-nilpotent singularities behave under Segre products, Veronese subrings, and the formation of diagonal hypersurface algebras. To accomplish these tasks, we introduce the generalized F-depth in analogy to Lyubeznik’s F-depth. These depth-like invariants track (generalized) weakly F-nilpotent singularities in a similar fashion as (generalized) depth tracks (generalized) Cohen-Macaulay singularities.

Keywords:

• Depth
• Frobenius
• graded
• local cohomology
• nilpotent

2020 Mathematics Subject Classification:

• 13A35
• 13D45
• 14B15

Acknowledgments

We are extremely grateful to Ian Aberbach, Luis Núñez-Betancourt, Alessandra Costantini, Hailong Dao, Jack Jeffries, Paolo Mantero, Vaibhav Pandey, Thomas Polstra, and Austyn Simpson for enlightening discussions from which the work here greatly benefited.

Footnotes

Notes

1 Graded in the sense of Section 5.

2 We use the word “generalized” here to indicate a finite length version of the condition considered as for generalized Cohen-Macaulay rings.

3 We do not spell this idea out carefully here, but the extensions to the bigraded case are routine.